Last updated: January 5, 2023

Calculating the moment of inertia for different shapes and cross sections is probably the only thing we used in structural engineering during our studies and later in our professional lives.

Although knowing how to derive and calculate the moment of inertia is very important, we will be able to use more and more formulas as we advance in our studies.

In this article, we show the most important and simple formulas for rectangular, I/H, circular and hollow circle cross sections, but also formulas with more steps for L, T and U shapes.

Everything is shown and explained using examples.

Now, before we get started, always remember that the unit of inertia is the fourth power of a unit length [$length^4$]. If you want to use $mm$ in your calculation, then the moment of inertia unit is $mm^4$.

But now let's get started.

## 1. Moment of Inertia - Rectangular Shape/Cross Section (Formula)

#### strong axis

$I_y = \frac{1}{12} \cdot h^3 \cdot w$

#### weak axis

$I_z = \frac{1}{12} \cdot h^3 \cdot w$

### sample calculation

alt = 240 mm, ancho = 120 mm

##### strong axis:

$I_y = \frac{1}{12} \cdot h^3 \cdot w = \frac{1}{12} \cdot (240 mm)^3 \cdot 120 mm = 1,3824 \cdot 10^8 mm ^4$

##### weak axis:

$I_z = \frac{1}{12} \cdot h \cdot w^3 = \frac{1}{12} \cdot 240 mm \cdot (120 mm)^3= 3,456 \cdot 10^7 mm^4 $

### Where is the moment of inertia of a rectangular cross section used in real designs?

- Calculation of structural bending stress of wooden beams (Here)
- Sheet metal (steel) structural stress calculation
- Calculation of structural stresses of concrete beams.

## 2. Moment of Inertia - Shape/Section I/H (formula)

#### strong axis

$I_y = \frac{w \cdot h^3}{12} – \frac{(w-t_w) \cdot (h-2\cdot t_f)^3}{12}$

#### weak axis

$I_z = \frac{(h-2 \cdot t_f) \cdot t_w^3}{12} + \frac{2 \cdot t_f \cdot w^3}{12}$

### sample calculation

$h$ = 300 mm, $w$ = 150 mm, $t_f$ = 10 mm, $t_w$ = 7 mm

##### strong axis:

$I_y = \frac{w \cdot h^3}{12} – \frac{(w-t_w) \cdot (h-2\cdot t_f)^3}{12} = \frac{150mm \cdot (300mm )^3}{12} – \frac{(150mm-7mm) \cdot (300mm-2\cdot 10mm)^3}{12} = 7,59 \cdot 10^7 mm^4$

##### weak axis:

$I_z = \frac{(h-2\cdot t_f) \cdot t_w^3}{12} + \frac{2\cdot t_f \cdot w^3}{12} = \frac{(300mm-2\cdot 10 mm) \cdot (7 mm)^3}{12} + \frac{2\cdot 10 mm \cdot (7 mm)^3}{12} = 5,63 \cdot 10^6 mm^4$

### Where is the moment of inertia of an I/H section used in real projects?

- Calculation of structural bending stress of wooden I-beams
- Calculation of structural bending stress of I/H steel columns and beams

## 3. Moment of Inertia - Circular Shape/Cross Section (Formula)

#### strong axis

$I_y = \frac{D^4 \cdot \pi}{64}$

#### weak axis

$I_z = \frac{D^4 \cdot \pi}{64}$

### sample calculation

Depth = 100mm

##### strong axis:

$I_y = \frac{D^4 \cdot \pi}{64} = \frac{(100mm)^4 \cdot \pi}{64} = 4,91 \cdot 10^6 mm^4$

##### weak axis:

$I_z = \frac{D^4 \cdot \pi}{64} = \frac{(100mm)^4 \cdot \pi}{64} = 4,91 \cdot 10^6 mm^4$

### Where is the moment of inertia of a circular section used in real designs?

- Tension rods for wind straps made of steel construction
- Structural concrete pillar

## 4. Moment of inertia - hollow round tube cross section (formula)

#### strong axis

$I_y = \frac{(D^4-d^4) \cdot \pi}{64}$

#### weak axis

$I_z = \frac{(D^4-d^4) \cdot \pi}{64}$

### sample calculation

Depth = 100 mm, Depth = 90 mm

##### strong axis:

$I_y = \frac{(D^4 – d^4) \cdot \pi}{64} = \frac{((100mm)^4 – (90mm)^4) \cdot \pi}{64} = 1.688 \cdot 10^6 mm^4$

##### weak axis:

$I_z = \frac{(D^4 – d^4) \cdot \pi}{64} = \frac{((100mm)^4 – (90mm)^4) \cdot \pi}{64} = 1.688 \cdot 10^6 mm^4$

### Where is the moment of inertia of a circular section used in real designs?

- Tension rods for wind straps made of steel construction
- steel columns

## 5. Moment of Inertia - Hollow Rectangular Piping Section (Formula)

#### strong axis

$I_y = \frac{W \cdot H^3 -w \cdot h^3}{12}$

#### weak axis

$I_z = \frac{W^3 \cdot H -w^3 \cdot h}{12}$

### sample calculation

Ancho = 120 mm, Alt = 240 mm, Ancho = 100 mm, Alt = 220 mm

##### strong axis:

$I_y = \frac{W \cdot H^3 – w \cdot h^3}{12} = \frac{120mm \cdot (240mm)^3 – 100mm \cdot (220mm)^3}{12}= 4 ,95 \cdot 10^7 mm^4$

##### weak axis:

$I_z = \frac{W^3 \cdot H – w^3 \cdot h}{12} = \frac{(120mm)^3 \cdot 240mm – (100mm)^3 \cdot 220mm}{12} = 1 ,62 \cdot 10^7 mm^4$

### Where is the moment of inertia of a circular section used in real designs?

- columns

## 6. Moment of Inertia - U Profile (Formula)

#### strong axis

$I_y = \frac{w \cdot h^3 -(w-t_w)\cdot (h-2t_f)^3}{12}$

##### Distance from point a to the center of gravity:

$y_{c} = \frac{1}{(h-2t_f) \cdot t_w + 2 \cdot w \cdot t_f} \cdot ((h-2t_f) \cdot t_w \cdot \frac{t_w}{2} + 2 \cdot w \cdot t_f \cdot \frac{w}{2})$

##### Moment of inertia - weak axis:

$I_z = \frac{(h-2 \cdot t_f) \cdot t_w^3}{12} + (h-2 \cdot t_f) \cdot t_w \cdot (y_c – \frac{t_w}{2})^ 2 + \frac{2 \cdot t_f \cdot w^3}{12} + 2 \cdot w \cdot t_f \cdot (\frac{w}{2} – y_c)^2$

### sample calculation

Anker = 100 mm, Alt = 80 mm, $t_f$ = 5 mm, $t_w$ = 5 mm

##### strong axis:

$I_y = \frac{w \cdot h^3 – (w – t_w) \cdot (h – 2t_f)^3}{12} = \frac{100mm \cdot (80mm)^3 – (100mm – 5mm) \ cdot (80mm – 2\cdot 5mm)^3}{12}= 1,55 \cdot 10^6 mm^4$

##### Distance from point a to the center of gravity:

$y_{c} = \frac{1}{(h-2t_f) \cdot t_w + 2 \cdot w \cdot t_f} \cdot ((h-2t_f) \cdot t_w \cdot \frac{t_w}{2} + 2 \cdot w \cdot t_f \cdot \frac{w}{2})$

$y_{c} = \frac{1}{(80mm-2\cdot 5mm) \cdot 5mm + 2 \cdot 100mm \cdot 5mm} \cdot ((80mm-2 \cdot 5mm) \cdot 5mm \cdot \frac {5mm}{2}$ $

+ 2 \cdot 100mm \cdot 5mm \cdot \frac{100mm}{2})$

$y_c = 37,69 mm$

##### weak axis:

$I_z = \frac{(h-2 \cdot t_f) \cdot t_w^3}{12} + (h-2 \cdot t_f) \cdot t_w \cdot (y_c – \frac{t_w}{2})^ 2 + \frac{2 \cdot t_f \cdot w^3}{12} + 2 \cdot w \cdot t_f \cdot (\frac{w}{2} – y_c)^2$

$I_z = \frac{(80mm-2 \cdot 5mm) \cdot (5mm)^3}{12} + (80mm-2 \cdot 5mm) \cdot 5mm \cdot (37,69mm – \frac{5mm}{ 2})^2$

$ + \frac{2 \cdot 5 mm \cdot (100 mm)^3}{12} + 2 \cdot 100 mm \cdot 5 mm \cdot (\frac{100 mm}{2} – 37,69 mm) ^2$

$I_z = 1,42 \cdot 10^6 mm^4$

### Where is the moment of inertia of a U-section used in real designs?

- Bracing steel or wood structures

## 7. Moment of Inertia - T Profile (Formula)

#### weak axis

$I_z = \frac{t_f \cdot w^3}{12} + \frac{h \cdot t_w^3}{12}$

##### Distance from point $z_c$ to center of gravity:

$z_{c} = \frac{1}{h \cdot t_w + w \cdot t_f} \cdot (h \cdot t_w \cdot \frac{h}{2} + w \cdot t_f \cdot (h + \ Bruch{t_f}{2})$

##### Moment of Inertia - Weak Axis:

$I_y = \frac{h^3 \cdot t_w}{12} + h \cdot t_w \cdot (\frac{h}{2} – z_c)^2 + \frac{t_f \cdot w^3}{12 } + w \cdot t_f \cdot (h + \frac{t_f}{2} – z_c)^2$

### sample calculation

ancho = 100 mm, Alt = 100 mm, $t_f$ = 5 mm, $t_w$ = 5 mm

##### Distance $z_c$ to center of gravity:

$z_{c} = \frac{1}{h \cdot t_w + w \cdot t_f} \cdot (h \cdot t_w \cdot \frac{h}{2} + w \cdot t_f \cdot (h + \ Bruch{t_f}{2})$

$z_{c} = \frac{1}{100mm \cdot 5mm + 100mm \cdot 5mm} \cdot (100mm \cdot 5mm \cdot \frac{100mm}{2} + 100mm \cdot 5mm \cdot (100mm + \ Bruch{5mm}{2})$

$z_c = 76,25 mm$

##### strong axis:

$I_y = \frac{h^3 \cdot t_w}{12} + h \cdot t_w \cdot (\frac{h}{2} – z_c)^2 + \frac{t_f \cdot w^3}{12 } + w \cdot t_f \cdot (h + \frac{t_f}{2} – z_c)^2 $

$I_y = \frac{(100 mm)^3 \cdot 5 mm}{12} + 100 mm \cdot 5 mm \cdot (\frac{100 mm}{2} – 76,25 mm)^2 + \frac {5 mm \cdot (100 mm) ^3}{12}$

$ + 100 mm \cdot 5 mm \cdot (100 mm + \frac{5mm}{2} – 76,25 mm)^2 $

$I_y = 1.107 \cdot 10^6 mm^4$

##### weak axis:

$I_z = \frac{t_f \cdot w^3}{12} + \frac{h \cdot t_w^3}{12}$

$I_z = \frac{5mm \cdot (100mm)^3}{12} + \frac{100mm \cdot (5mm)^3}{12}$

$I_z = 4.177 \cdot 10^5 mm^4$

## 8. Moment of Inertia - Asymmetric I/H Profile (Formula)

#### weak axis

$I_z = \frac{t_{f.t} \cdot w_{t}^3}{12} + \frac{(h-t_{f.t}-t_{f.b})\cdot t_w^3}{12} +\ frac{t_{f.b} \cdot w_{b}^3}{12}$

##### Distance from point $z_c$ to center of gravity:

$z_c=(\frac{1}{w_t \cdot t_{f.t}+w_b \cdot t_{f.b}+(h-t_{f.t}-t_{f.b}) \cdot t_w}) \cdot (w_t \cdot t_{f.t} \cdot \frac{t_{f.t}}{2}+(h-t_{f.t}-t_{f.b}) \cdot t_w \cdot(t_{f.t}+\frac{(h-t_{ f.t}-t_{f.b}}{2})$

$+w_b \cdot t_{f.b} \cdot(h-\frac{t_{f.b}}{2}))$

##### Moment of Inertia - Weak Axis:

$I_y=\frac{w_t \cdot t_{f.t}^3}{12}+w_t \cdot t_{f.t} \cdot(z_c-\frac{t_{f.t}}{2})^2+\frac{ t_w \cdot(h-t_{f.t}-t_{f.b})^3}{12}$

$+t_w \cdot(h-t_{f.t}-t_{f.b}) \cdot(z_c-(t_{f.t}+\frac{(h-t_{f.t}-t_{f.b})}{2}) )^2+\frac{w_b \cdot t_{f.b}^3}{12}+w_b \cdot t_{f.b} \cdot(z_c-h-\frac{t_{f.b}}{2})^2$

### sample calculation

$w_t = 200 mm$, $w_b = 100 mm$, $h = 200 mm$, $t_{f.t} = 20 mm$, $t_{f.b} = 10 mm$, $t_w = 10 mm$

##### Distance $z_c$ to center of gravity:

$z_c=(\frac{1}{200mm \cdot 20mm+100mm \cdot 10mm+(200mm-20mm-10mm) \cdot 10mm}) \cdot$

$(200mm \cdot 20mm \cdot \frac{20mm}{2}+(200mm-20mm-10mm) \cdot 10mm \cdot(20mm+\frac{200mm-20mm-10mm}{2})$

$+100mm \cdot 10mm \cdot(200mm-\frac{10mm}{2})) = 61,72mm$

##### strong axis:

$I_y=\frac{200mm \cdot (20mm)^3}{12}+200mm \cdot 20mm \cdot(61.72mm-\frac{20mm}{2})^2+\frac{10mm \cdot(200mm- 20 mm-10 mm)^3}{12}$

$+10 mm \cdot(200 mm-20 mm-10 mm) \cdot(61,72 mm-(20 mm+\frac{(200 mm-20 mm-10 mm)}{2}))^2$

$+\frac{100mm \cdot (10mm)^3}{12}+100mm \cdot 10mm \cdot(61,72mm-200mm-\frac{10mm}{2})^2 = 3,865 \cdot 10^7 mm ^ 4 $

##### weak axis:

$I_z = \frac{20mm \cdot (200mm)^3}{12} + \frac{(200mm-20mm-10mm)\cdot (10mm)^3}{12} +\frac{10mm \cdot (100mm) ^3}{12} = 1,418 \cdot 10^7 mm^4$

If you're new to structural design, check out our design tutorials where you can learn how to use the moment of inertia to design design elements such as beams

- Wooden ceiling beams design.
- Buckling design of wooden pillars.
- Neck beam buckling design

Are you missing a moment of inertia formula for a shape or cross section that we missed in this article? Tell us in the comments✍️

## FAQs

### How do you find the moment of inertia of different shapes? ›

Generally, **for uniform objects, the moment of inertia is calculated by taking the square of its distance from the axis of rotation (r ^{2}) and the product of its mass**. Now, in the case of non-uniform objects, we can calculate the moment of inertia by taking the sum of individual point masses at each different radius.

**What is the structure of the formula for moment of inertia? ›**

The formula for the moment of inertia is the “sum of the product of mass” of each particle with the “square of its distance from the axis of the rotation”. The formula of Moment of Inertia is expressed as **I = Σ m _{i}r_{i}^{2}**.

**What are the formulas for moment of inertia for different shapes Class 11? ›**

Moment of Inertia Formula

m = Sum of the product of the mass. r = Distance from the axis of the rotation. ⇒ The dimensional formula of the moment of inertia is given by, **M ^{1} L^{2} T^{0}**. The role of the moment of inertia is the same as the role of mass in linear motion.

**What formula is used in the experimental determination of the moment of inertia? ›**

By applying a known torque to a rigid body, measuring the angular acceleration, and using the relationship **τ = Iα**, the moment of inertia can be determined.

**Why is moment of inertia different for different shapes? ›**

The moment of inertia of an object depends on its mass, size, and shape. **increases as the mass is more concentrated toward the outside of the object**. Thus objects of equal masses and even equal radii can have different moments of inertia.

**What are the formulas for moment of inertia for different 3D shapes? ›**

Shape moment of inertia for 3D shapes

The moment of inertia **I=∫r ^{2}dm** for a hoop, disk, cylinder, box, plate, rod, and spherical shell or solid can be found from this figure.

**What is the formula for moment of inertia for a square or rectangular section? ›**

To sum up, the formula for finding the moment of inertia of a rectangle is given by **I=bd³ ⁄ 3**, when the axis of rotation is at the base of the rectangle.

**What is the formula for moment of inertia of a solid sphere? ›**

**I=ICM+MX2**, where I is the moment of inertia about axis passing through O, where ICM is the moment of inertia of the solid sphere having radius R about an axis parallel to the center of gravity, M is the mass of the body and X is the distance between the axes.

**What is use of moment of inertia in analysis of structures? ›**

Moment of Inertia is an important geometric property used in structural engineering, as it is **directly related to the amount of material strength your section has**. Generally speaking, the higher the moment of inertia, the more strength it has and the less it will deflect under load.

**What are the different types of moment of inertia? ›**

The three main types of the moment of force are the **mass moment of inertia, the area moment of inertia, and the polar moment of inertia**.

### What is the formula in solving for the moment of inertia of a multiple object system? ›

How to Calculate Moment of Inertia for Multiple Objects around an Axis. Step 1: For each object, identify its mass and distance to the axis of rotation. Step 2: Use the formula **I=∑mr2** I = ∑ m r 2 to calculate the moment of inertia.

**How do you solve the moment formula? ›**

**Moment of force = F x d**

F is the force applied, d is the distance from the fixed axis, Moment of force is expressed in newton meter (Nm). Moment of force formula can be applied to calculate the moment of force for balanced as well as unbalanced forces.

**What is moment of inertia and how it is calculated? ›**

Moments of inertia can be found by summing or integrating over every 'piece of mass' that makes up an object, multiplied by the square of the distance of each 'piece of mass' to the axis. In integral form the moment of inertia is **I=∫r2dm** I = ∫ r 2 d m .

**What is moment of inertia explained simply? ›**

The moment of inertia is **a physical quantity which describes how easily a body can be rotated about a given axis**. It is a rotational analogue of mass, which describes an object's resistance to translational motion. Inertia is the property of matter which resists change in its state of motion.

**What are the two theorems of moment of inertia? ›**

Solution : Two theorems of moment of inertia are **theorem of parallel axes and theorem of perpendicular axes**.

**On what factors does moment of inertia depend? ›**

Moment of inertia of a body depends on **position and orientation of the axis of rotation**. It also depends on shape, size of the body and also on the distribution of mass of the body about the given axis.

**Which shape has the greatest moment of inertia? ›**

From smallest to largest moment of inertia, and thus first to last to reach the bottom: **sphere, cylinder, hoop**.

**What does the moment of inertia depend on? ›**

The moment of inertia of an object usually depends on **the direction of the axis**, and always depends on the perpendicular distance from the axis to the object's centre of mass.

**How do you find the moment of inertia of a 2d shape? ›**

= **∫ (dy)2 dA + 2 * ∫ (dy*y') dA + ∫ (y')2 dA**.

**What is the SI unit and dimensional formula of moment of inertia? ›**

Or, **MOI = [M ^{1} L^{0} T^{0}] × [M^{0} L^{1} T^{0}]^{2} = M^{1} L^{2} T^{0}**. Therefore, the moment of inertia is dimensionally represented as M

^{1}L

^{2}T

^{0}.

### What is the formula for moment of inertia of circular cross section? ›

Moment Of Inertia Of A Circle

This equation is equivalent to **I = π D ^{4} / 64** when we express it taking the diameter (D) of the circle.

**What is the moment of inertia for a rectangular beam? ›**

What is the moment of inertia of a rectangle? In the case where the axis passes through the centroid, the moment of inertia of a rectangle is given as **I = bh3 / 12**.

**What is the moment of inertia of a solid square? ›**

Moment of inertia of a solid about its geometrical axis is given by **l=52MR2** where M is mass & R is radius.

**What is moment of inertia of a solid and hollow sphere? ›**

From the above table, it is clear that the moment of inertia of a solid sphere of mass 'm' and radius 'R' about an axis passing through the center is: **I 1 = 2 5 m r 2** . Moment of inertia of a Hollow sphere of mass 'm' and radius 'R' about an axis passing through the center is: I 2 = 2 3 m r 2 .

**What is the moment of inertia of a circle? ›**

The moment of inertia of a circle, also known as the second-moment area of a circle, is commonly calculated using the formula **I = R4 / 4**. The radius is R, and the axis passes through the centre. When we represent this equation in terms of the circle's diameter (D), it becomes I = D4 / 64.

**How do you calculate moment in structural analysis? ›**

Calculate the unbalanced moment at each joint and distribute the same to the ends of members connected at that joint. Carry over one-half of the distributed moment to the other ends of members. Add or subtract these latter moments (moments obtained in steps three and four) to or from the original fixed-end moments.

**What are the 3 types of inertia explain with example? ›**

Inertia of rest: The inability of a body to change by itself its state of rest is called inertia of rest. Inertia of direction: The inability of a body to change by itself its direction of motion. Inertia of motion: The inability of the body to change by itself its state of motion is called inertia of motion.

**What are the three types of inertia answer? ›**

- Inertia:
- Different types of inertia with examples are given below.
- Inertia of rest:
- Inertia of motion:
- Inertia of direction:

**How many types of inertia explain with example? ›**

It is of **three types**: The inertia of rest: Tendency of a body to remain in the state of rest. The inertia of direction: Tendency of a body to remain in a particular direction. The inertia of motion: Tendency of a body to remain in a state of uniform motion.

**What is the relation between moment of inertia and radius of gyration? ›**

The radius of gyration is the distance from an axis of a body to the point in the body whose **moment of inertia is equal to the moment of inertia of the entire body**. The radius of gyration is equal to the square root of the ratio between the moment of inertia of the entire body and the mass of the whole system.

### What is the moment of inertia of a combined system? ›

The moment of inertia of a rigid composite system is **the sum of the moments of inertia of its component subsystems** (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.

**How do you find the moment of multiple forces? ›**

If different forces act at different points then the total moment about any point P is the algebraic (i.e. including the signs) sum of each moment about P. **R = F1 + F2** .

**Why do we calculate moments? ›**

Moments in mathematical statistics involve a basic calculation. These calculations can be used **to find a probability distribution's mean, variance, and skewness**. Using this formula requires us to be careful with our order of operations.

**What is the use of three moment equation? ›**

The three moment equation **expresses the relation between bending moments at three successive supports of a continuous beam, subject to a loading on a two adjacent span with or without settlement of the supports**.

**What is moment of inertia with example? ›**

Real life examples of moment of inertia

FLYWHEEL of an automobile: Flywheel is a heavy mass mounted on the crankshaft of an engine. The magnitude of MOI of the flywheel is very high and helps in storing the energy. Hollow shaft- An hollow shaft transmits more power compared to that of a solid shaft(both of same mass).

**Why do we calculate mass moment of inertia? ›**

To sum up, the mass moment of inertia clearly **indicates what degree of resistance is offered by a body to rotational acceleration about an axis**. If a body has a large mass moment of inertia it can be said or deduced that it offers high resistance to angular acceleration.

**How do you calculate inertia force? ›**

The law states that **F=ma** F = m a , where F is the force applied to an object, m is the inertial mass of the object, and a is the accleration of the object.

**What is the law of moment of inertia? ›**

law of inertia, also called Newton's first law, postulate in physics that, **if a body is at rest or moving at a constant speed in a straight line, it will remain at rest or keep moving in a straight line at constant speed unless it is acted upon by a force**.

**What is the difference between moment and moment of inertia? ›**

**Momentum is a vector quantity as it is the tendency of a body to remain in motion.** Inertia is a scalar quantity as it is the resistance offered by the body to any change in its velocity.

**Why is it called moment of inertia? ›**

Moment of inertia **resists rotational motion** and hence called moment of inertia and not moment of force.

### What is moment of inertia Ixx and Iyy? ›

The quantities Ixx, Iyy, and Izz are called moments of inertia with respect to the x, y and z axis, respectively, and are given by. **Ixx = ∫m (y′2 + z′2) dm , Iyy = ∫m (x′2 + z′2) dm , Izz = ∫m (x′2 + y′2) dm** .

**How do you find the moment of inertia of two objects? ›**

How to Calculate Moment of Inertia for Multiple Objects around an Axis. Step 1: For each object, identify its mass and distance to the axis of rotation. Step 2: Use the formula **I=∑mr2 I = ∑ m r 2** to calculate the moment of inertia.

**What is the moment of inertia for a 2d rectangle? ›**

(10.2. 2) 2) **I x = b h 3 3** . This is the formula for the moment of inertia of a rectangle about an axis passing through its base, and is worth remembering.

**How do you find the moment of inertia of a composite section? ›**

If a composite part has an empty region(hole),its moment of inertia is found by **subtracting the moment of inertia of this region from the moment of inertia of the entire part including the region**. Examples - Example(6): Determine the moment of inertia of the beam cross sectional area about the x axis.

**How do you find the moment of inertia with multiple masses? ›**

**Summary**

- Moments of inertia can be found by summing or integrating over every 'piece of mass' that makes up an object, multiplied by the square of the distance of each 'piece of mass' to the axis. ...
- Moment of inertia is larger when an object's mass is farther from the axis of rotation.

**How do you find the moment of inertia and bond length? ›**

**We can calculate the bond length from the moment of inertia and this reduced mass:**

- r = √ I.
- µ = √
- √ √ √
- (3.348 × 10−47 kgm2) ( 0.995116912.

**How do you find the moment of inertia of a flat plate? ›**

To sum up, the formula for finding the moment of inertia of a rectangular plate about its center is given by **I = mb³/12**, when the axis of rotation passes through the center of the rectangular plate and also parallel to the edge.

**What is the moment of inertia of rod perpendicular to axis? ›**

The moment of inertia of a rod about an axis through its centre and perpendicular to it is **121ML2** (where M is the mass and L, the length of the rod).

**How do you find the moment of inertia of an equilateral triangle? ›**

**$I = \Sigma mr^2$** where m is the mass of the rotating body with respect to the distance r of the mass from the axis of rotation.

**What is polar moment of inertia for circular and square section? ›**

Polar moment of inertia basically describes the cylindrical object's (including its segments) resistance to torsional deformation when torque is applied in a plane that is parallel to the cross-section area or in a plane that is perpendicular to the object's central axis.

### How do you find the polar moment of inertia for a circular cross-section? ›

For a solid circular section, use the polar moment of inertia formula **J = πR⁴/2**, where R is the radius, and J is the polar moment of inertia.